My BIL & SIL bought their son a piano when they lived in a block of small, new build flats - the kind with no noise insulation at all.
We did tell them that the neighbours might not be too thrilled but apparently the child was such an amazing piano player that the neighbours should be grateful to be allowed to hear him. I mean, he's ok but no one needs to hear 3 hours of practice and scales every day.
Anyway, my point, if there is one, is yes, they are BU, and you are a saint. I would definitely put a note through the door, I'd try to keep it civil.
We met our in-laws neighbour once, DH said something about the piano & I swear I saw the neighbours BP rise about 40 points. The vein in his neck stood out and he clenched his fists and teeth. Poor bloke. They must have had a party when the in-laws moved out.